任意维数薄凸域上Neumann特征值的二次比较
A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion
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中文总结 AI 辅助
本文研究任意维数薄凸域的Neumann特征值，证明其与加权线段谱的$O(\varepsilon^2)$比较，并指出该二次精度最优。
中文摘要 AI 辅助
设$\Omega\subset\mathbb R^n$为有界凸域，其围绕所选直径线段呈薄形。我们将其Neumann谱与该线段以其垂直截面的$(n-1)$维体积加权的谱进行比较。我们证明了均值零逆算子的$O(\varepsilon^2)$比较，并因此对每个固定指标在任意维数$n\ge2$下证明了$O(\varepsilon^2)$特征值比较。常数仅依赖于维数和特征值指标。薄矩形表明二次指数是最优的。
英文摘要
Let $Ω\subset\mathbb R^n$ be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the $(n-1)$-dimensional volumes of its perpendicular sections. We prove an $O(\varepsilon^2)$ comparison of the mean-zero inverse operators and, consequently, an $O(\varepsilon^2)$ eigenvalue comparison for every fixed index in every dimension $n\ge2$. The constants depend only on the dimension and the eigenvalue index. Thin rectangles show that the quadratic exponent is optimal.
发表机构
- Shantou University（汕头大学）
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